Problem: $ C = \left[\begin{array}{rrr}2 & -2 & 0 \\ 3 & 0 & 0 \\ -1 & -2 & 0\end{array}\right]$ $ E = \left[\begin{array}{rr}3 & 0 \\ -2 & 1\end{array}\right]$ Is $ C E$ defined?
Solution: In order for multiplication of two matrices to be defined, the two inner dimensions must be equal. If the two matrices have dimensions $( m \times  n)$ and $( p \times q)$ , then $ n$ (number of columns in the first matrix) must equal $ p$ (number of rows in the second matrix) for their product to be defined. How many columns does the first matrix, $ C$ , have? How many rows does the second matrix, $ E$ , have? Since $ C$ has a different number of columns (3) than $ E$ has rows (2), $ C E$ is not defined.